A local hybrid kernel meshless method for numerical solutions of two‐dimensional fractional cable equation in neuronal dynamics

Abstract

This study deals with obtaining numerical solutions of two-dimensional (2D) fractional cable equation in neuronal dynamics by using a recently introduced meshless method. In solution process at first stage, time derivatives that are appeared in the considered problem are discretized by using finite difference method. Then a meshless method based on hybridization of Gaussian and cubic kernels is developed in local fashion. The problem is solved both on regular and irregular domians. L ∞ and RMS error norms are calculated and compared with other numerical methods in literature as well as exact solutions. Also, obtained condition numbers are monitored. Numerical simulations show that local hybrid kernel meshless method is a thriving method for solving 2D fractional cable equation on regular and irregular domians. KEYWORDS 2D fractional cable equation, irregular domain, local hybrid kernel meshless method 1 INTRODUCTION Fractional calculus and fractional differential equations have attracted more attention recently due to their ability of handling many problems from various branches of physics, engineering, and chemistry in a more realistic approach [1, 2]. It has been recognized that fractional derivative has more superiorities than integer order derivative. For instance fractional derivatives can be used to describe memory and hereditary properties of physical materials [3-5]. This feature was not identified in integer order derivatives. There are a lot of applications of fractional calculus and fractional partial differential equations (PEDs) in different branches of science such as viscoelasticity, physics, chemistry, fluid mechanics, control, continuous-time random walks, and finance [6]. Some classic books about fractional calculus and its applications are [7-10].

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

“…A meshless numerical scheme with rigorous error analysis for the two-dimensional CO FCE was reported in [42]. Oruç [43] suggested a local hybrid kernel meshless method to deal with the two-dimensional CO FCE. Zheng et al [44] utilized a finite difference scheme in the temporal direction and a Legendre spectral method in the spatial direction to solve the two-dimensional distributed-order FCE.…”

A new implicit high-order iterative scheme for the numerical simulation of the two-dimensional time fractional Cable equation

“…Bhrawy and Zaky 22 used spectral collocation method for both 1-D and 2-D FCE which is based on shifted Jacobi collocation method combined with the Jacobi operational matrix for fractional derivative. Ömer 23 discussed the numerical solution for 2-D FCE using a meshless numerical method which is based on the hybridization of Gaussian and cubic kernels. Moreover, the FCE is solved on both regular and irregular domains.…”

Analysis of Two-Level Mesh Method for Partial Integro-Differential Equation

“…In recent years, there are many researchers that studied the meshless method for solving partial differential equations. For example, Oruç developed a meshless approach based on radial basis function-finite difference (RBF-FD) method for solving one-dimensional, two-dimensional, and three-dimensional Schrödinger system [24,25], two-dimensional fractional cable equation [26], and wave equations [27,28]. Nikan and Avazzadeh [29] constructed an improved localized radial basis-pseudospectral numerical method for solving fractional reaction-subdiffusion equation.…”